Multiple Solutions for the Asymptotically Linear Kirchhoff Type Equations on RN

Abstract

The multiplicity of positive solutions for Kirchhoff type equations depending on a nonnegative parameter $\lambda$ on ${\mathbb{R}}^N$ is proved by using variational method. We will show that if the nonlinearities are asymptotically linear at infinity and $\lambda >\mathrm{0}$ is sufficiently small, the Kirchhoff type equations have at least two positive solutions. For the perturbed problem, we give the result of existence of three positive solutions.